Geometric Complexity Theory IV: Nonstandard Quantum Group for the Kronecker Problem

Introduction Basic concepts and notation Hecke algebras and canonical bases The quantum group $GL_q(V)$ Bases for $GL_q(V)$ modules Quantum Schur-Weyl duality and canonical bases Notation for $GL_q(V) \times GL_q(W)$ The nonstandard coordinate algebra $\mathscr{O}(M_q(\check{X}))$ Nonstandard determinant and minors The nonstandard quantum groups $GL_q(\check{X})$ and $\texttt{U}_q(\check{X})$ The nonstandard Hecke algebra $\check{\mathscr{H}}_r$ Nonstandard Schur-Weyl duality Nonstandard representation theory in the two-row case A canonical basis for $\check{Y}_\alpha$ A global crystal basis for two-row Kronecker coefficients Straightened NST and semistandard tableaux} A Kronecker graphical calculus and applications Explicit formulae for Kronecker coefficients Future work Appendix A. Reduction system for ${\mathscr{O}}(M_q(\check{X}))$ Appendix B. The Hopf algebra ${\mathscr{O}}_{q}^\tau$ Bibliography

[1]  K. V. Subrahmanyam,et al.  Quantum deformations of the restriction of GLmn(C)-modules to GLm(C) × GLn(C) , 2009, ArXiv.

[2]  Emmanuel Briand,et al.  Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes (extended abstract) , 2008 .

[3]  K. V. Subrahmanyam,et al.  A geometric approach to the Kronecker problem I: The two row case , 2008 .

[4]  J. Blasiak.  $W$-graph versions of tensoring with the $\S_n$ defining representation , 2008, 0809.4810.

[5]  Mike Zabrocki,et al.  Expressions for Catalan Kronecker Products , 2008, 0809.3469.

[6]  Ketan Mulmuley,et al.  Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties , 2006, SIAM J. Comput..

[7]  Eric C. Rowell,et al.  An algebra-level version of a link-polynomial identity of Lickorish , 2006, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  Ketan Mulmuley,et al.  Geometric Complexity Theory VIII: On canonical bases for the nonstandard quantum groups , 2007, ArXiv.

[9]  Ketan Mulmuley,et al.  Geometric Complexity Theory VII: Nonstandard quantum group for the plethysm problem , 2007, ArXiv.

[10]  Ketan Mulmuley,et al.  Geometric Complexity Theory IV: quantum group for the Kronecker problem , 2007, ArXiv.

[11]  J. Brundan.  Dual canonical bases and Kazhdan–Lusztig polynomials☆ , 2005, math/0509700.

[12]  Rosa C. Orellana,et al.  On the Kronecker Product s(n-p,p) * sλ , 2005, Electron. J. Comb..

[13]  Arkady Berenstein,et al.  Braided symmetric and exterior algebras , 2005, math/0504155.

[14]  Ketan Mulmuley,et al.  Geometric Complexity III: on deciding positivity of Littlewood-Richardson coefficients , 2005, ArXiv.

[15]  J. Landsberg,et al.  On the Ideals of Secant Varieties of Segre Varieties , 2003, Found. Comput. Math..

[16]  S. Bravyi.  Requirements for copatibility between local and multipartite quantum states , 2003, Quantum Inf. Comput..

[17]  J. Weyman.  Cohomology of Vector Bundles and Syzygies , 2003 .

[18]  Jin Hong,et al.  Introduction to Quantum Groups and Crystal Bases , 2002 .

[19]  Ketan Mulmuley,et al.  Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems , 2002, SIAM J. Comput..

[20]  A. Sudbery.  QUANTUM GROUPS AND THEIR REPRESENTATIONS (Texts and Monographs in Physics) , 2000 .

[21]  S. Majid.  Quantum groups and noncommutative geometry , 2000, hep-th/0006167.

[22]  M. Rosas.  The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes , 2000, math/0001084.

[23]  K. Schmüdgen,et al.  Quantum Groups and Their Representations , 1998 .

[24]  G. Lehrer,et al.  Cellular algebras , 1996 .

[25]  Arun Ram.  Seminormal Representations of Weyl Groups and Iwahori‐Hecke Algebras , 1995, math/9511223.

[26]  Jeffrey B. Remmel,et al.  On the Kronecker product of Schur functions of two row shapes , 1994 .

[27]  Toshiki Nakashima,et al.  Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras , 1993 .

[28]  Masatoshi Noumi,et al.  Finite dimensional representations of the quantum group GL_q(n;C) and the zonal spherical functions on U_q(n-1)\U_q(n) , 1993 .

[29]  Masaki Kashiwara,et al.  Global crystal bases of quantum groups , 1993 .

[30]  George Lusztig,et al.  Introduction to Quantum Groups , 1993 .

[31]  Susan Montgomery,et al.  Hopf algebras and their actions on rings , 1993 .

[32]  Takahiro Hayashi.  Non-Existence of Homomorphisms between Quantum Groups , 1992 .

[33]  G. Lusztig,et al.  Canonical bases in tensor products. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[34]  Arun Ram,et al.  A Frobenius formula for the characters of the Hecke algebras , 1991 .

[35]  William F. Schelter,et al.  Quantum deformations of GLn , 1991 .

[36]  R. Carter.  REFLECTION GROUPS AND COXETER GROUPS (Cambridge Studies in Advanced Mathematics 29) , 1991 .

[37]  M. Kashiwara,et al.  On crystal bases of the $Q$-analogue of universal enveloping algebras , 1991 .

[38]  N. Vilenkin,et al.  Representation of Lie groups and special functions , 1991 .

[39]  Nicolai Reshetikhin,et al.  Multiparameter quantum groups and twisted quasitriangular Hopf algebras , 1990 .

[40]  Masaki Kashiwara,et al.  Crystalizing theq-analogue of universal enveloping algebras , 1990 .

[41]  A. Sudbery.  Consistent multiparameter quantisation of GL(n) , 1990 .

[42]  George Lusztig,et al.  Canonical bases arising from quantized enveloping algebras , 1990 .

[43]  Jeffrey B. Remmel,et al.  A formula for the Kronecker products of Schur functions of hook shapes , 1989 .

[44]  S. Woronowicz,et al.  Compact matrix pseudogroups , 1987 .

[45]  Michio Jimbo,et al.  A q-analogue of U(g[(N+1)), Hecke algebra, and the Yang-Baxter equation , 1986 .

[46]  David A. Buchsbaum,et al.  Schur Functors and Schur Complexes , 1982 .

[47]  C. Curtis,et al.  Methods of representation theory--with applications to finite groups and orders , 1981 .

[48]  Irving Reiner,et al.  Methods of Representation Theory , 1981 .

[49]  A. Lascoux.  Produit de kronecker des representations du groupe symetrique , 1980 .

[50]  D. Kazhdan,et al.  Representations of Coxeter groups and Hecke algebras , 1979 .

[51]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .